Abstract

Recently, Takahashi and Takahashi proposed an iterative algorithm for solving
a problem for finding common solutions of generalized equilibrium problems governed by
inverse strongly monotone mappings and of fixed point problems for nonexpansive mappings.
In this paper, we provide a result that allows for the removal of one condition ensuring the
strong convergence of the algorithm.

1. Introduction

Let be a real Hilbert space and a nonempty closed convex subset. A generalized equilibrium problem is formulated as a problem of finding a point with the property
where is a bifunction and is a nonlinear mapping. In particular, if is the zero mapping, then problem (1) is reduced to an equilibrium problem; find a point with the property
We will denote by and the solution set of problem (1) and problem (2), respectively. A fixed point problem (FPP) is to find a point with the property
where is a nonlinear mapping. The set of fixed points of is denoted as .

The problem under consideration in this paper is to find a common solution of problem (1) and of FPP (3). Namely, we seek a point such that
We consider problem (4) in the case whenever is a -inverse strongly monotone mapping and is a nonexpansive mapping. To solve problem (4), Takahashi and Takahashi [1] introduced an algorithm which generates a sequence by the iterative procedure
where , and are chosen so that
Under these conditions, they proved that the sequence generated by (5) can be strongly convergent to a solution of problem (4).

It is the aim of this paper to continue the study of algorithm (5). We will show that problem (4) is in fact a special fixed point problem for a nonexpansive mapping (a composition of a nonexpansive mapping and an averaged mapping). Our approach mainly uses the properties of averaged mappings, which is different from the existing methods invented by Takahashi and Takahashi. Moreover, we shall prove that condition sufficient to guarantee the convergence of algorithm (5) is superfluous.

2. Preliminaries and Notations

Notation 1. strong convergence, weak convergence and the set of the weak cluster points of .Denote by the projection from onto ; namely, for , is the unique point in with the property
It is well known that is characterized by the inequality
We will use the following notions on nonlinear mappings . (i)is nonexpansive if
(ii) is firmly nonexpansive if
(iii) is -averaged if there exist a constant and a nonexpansive mapping such that , where is the identity mapping on . (iv) is -inverse strongly monotone if there is a constant such that
The next lemma is referred to as the demiclosedness principle for nonexpansive mappings (see [2]).

Lemma 1. Let be a nonempty closed convex subset of and a nonexpansive mapping with . If is a sequence in such that and , then ; that is, .

Averaged mappings will play important role in our convergence analysis. We therefore collect some useful properties of averaged mappings (see, e.g., [3–5]).

Lemma 2. The following assertions hold. (i) is firmly nonexpansive if and only if is -averaged.(ii)If is -averaged, , then is -averaged. (iii)If is -averaged, then for any and for all ,

From now on, we assume that is a bifunction so that (A1), for all ; (A2) is monotone; that is, , for all ; (A3), for all ; (A4) for each is convex and lower semicontinuous. Under these assumptions, the following results hold (see [6, 7]).

Lemma 3. Let satisfy (A1)–(A4). Then for any and , there exists so that
Moreover if , then (i) is single valued and ;(ii) is firmly nonexpansive; (iii) is closed and convex.

Lemma 4. Let be a nonnegative real sequence satisfying
where and are real sequences. Then provided that (i), ;(ii) or .

3. Algorithm and Its Convergence

We begin with the following lemma.

Lemma 5. Assume that is -inverse strongly monotone mapping for some . Given a real number such that , set with defined as in Lemma 3. Then the following assertions hold: (a) is single valued and ; (b) is -averaged; (c)given , it follows that
(d)if , then for all

Proof. (a) It is readily seen that is single valued because is single valued. The equality follows from the definition of .(b) It follows that
Since is -inverse strongly monotone, is nonexpansive. Observe that
which implies that is -averaged. Consequently (b) follows from part (ii) of Lemma 2 and (c) follows from part (iii) of Lemma 2.(d) Let and . By definition of ,
Letting in (19) yields
Similarly,
Adding up these inequalities and using the monotonicity of ,
or equivalently,
Hence, . By the triangle inequality,
which is the result as desired.

For every , if we define , where is defined as in Lemma 3, then we can rewrite algorithm (5) as

Theorem 6. Let be a bifunction satisfying (A1)–(A4), a -inverse strongly monotone mapping for some , and a nonexpansive mapping so that the solution set is nonempty. If the following conditions hold:
then the sequence generated by (25) converges strongly to .

Before proving the theorem, we need some lemmas.

Lemma 7. Let the conditions in Theorem 6 be satisfied. If and are the sequences generated by (25), then both and are bounded.

Proof. Let be fixed. We have
on the other hand,
Altogether
By induction, is bounded and so is .

Lemma 8. Let the conditions in Theorem 6 be satisfied. If and , then and .

Proof. Let . By part (d) of Lemma 5,
Since is nonexpansive, applying the demiclosedness principle yields
On the other hand, we see that
which implies that
Using again the demiclosedness principle gets the desired result.

Proof of Theorem 6. Let . Using Lemma 5(c), we have
By the subdifferential inequality,
which implies that
By our assumption, there exists so that for all ,
and . Consequently,
Set , and let be a subsequence so that it includes all elements in with the property; each of them is less than or equal to the term after it. Following an idea developed by Maingé [9], we next consider two possible cases on .Case 1. Assume that is finite. Then there exists so that for all , and therefore must be convergent. It follows from (38) that
where is a sufficiently large real number. Consequently, both and converge to zero, and by Lemma 8 we conclude that and . Hence,
where the inequality uses (8). It then follows from (38) that
We therefore apply Lemma 4 to conclude that .Case 2. Assume now that is infinite. Let be fixed. Then there exists so that . By the choice of , we see that is the largest one among ; in particular
Then we deduce from (38) that
Applying Lemma 8 yields and . Similarly
It follows again from (38) and inequality (42) that
Taking in (44) yields
Moreover, we deduce from algorithm (25) that
which together with (43) implies that . Consequently immediately follows from (42).

4. Applications

In this section we present several applications. First we consider a problem for finding a common solution of equilibrium problem (2) and fixed point problem (3); namely, find so that
Taking in Theorem 6 and noting that zero mapping is -inverse strongly monotone for any positive number , one can easily get the following.

Corollary 9. Let be a bifunction satisfying (A1)–(A4) and a nonexpansive mapping so that the solution set of problem (48) is nonempty. Given , let generated by the iterative algorithm:
If the following conditions hold:
then the sequence converges strongly to a solution of problem (48).

A variational inequality problem (VIP) is formulated as a problem of finding a point with the property
We will denote the solution set of VIP (51) by . Next we consider a problem for finding a common solution of variational inequality problem (51) and of fixed point problem (3), namely; find so that
Taking in (1), we note that the generalized equilibrium problem is reduced to the variational problem (51). Thus applying Theorem 6 gets the following.

Corollary 10. Let be -inverse strongly monotone mapping and a nonexpansive mapping so that the solution set of problem (52) is nonempty. Given , let generated by the iterative algorithm:
If the following conditions hold:
then the sequence converges strongly to a solution of problem (52).

Consider the optimization problem of finding a point with the property
where is a convex and differentiable function. We say that the differential is -Lipschitz continuous, if
Denote by the solution set of problem (55). Finally we consider a problem for finding a common solution of optimization problem (55) and of fixed point problem (3), namely; find so that
By [10, Lemma 5.13], problem (55) is equivalent to the variational inequality problem
Taking in Corollary 10, we have the following result.

Corollary 11. Let be a convex and differentiable function so that is -Lipschitz continuous. Let be a nonexpansive mapping so that the solution set of problem (57) is nonempty. Given , let generated by
If the following conditions hold:
then the sequence converges strongly to a solution of problem (57).

Proof. It suffices to note that if is -Lipschitz continuous, then it is -inverse strongly monotone mapping (see [11, Corollary 10]). Consequently Corollary 10 applies and the result immediately follows.

Remark 12. We can further apply the previous method to find a common solution for fixed point and split feasibility problems, as well as for fixed point and convex constrained linear inverse problems (see [12]).

Acknowledgment

This work is supported by the National Natural Science Foundation of China, Tianyuan Foundation (11226227).